Problem A. 394. (February 2006)
A. 394. Let a_{1},a_{2},...,a_{N} be nonnegative reals, not all 0. Prove that there exists a sequence 1=n_{0}<n_{1}<...<n_{k}=N+1 of integers such that
n_{1}a_{n0}+n_{2}a_{n1}+...+n_{k}a_{nk1}<3(a_{1}+a_{2}+...+a_{N}).
(5 pont)
Deadline expired on March 16, 2006.
Solution. Define a_{N+2}+a_{N+3}=...=0 as well and look for an infinite sequence 1=n_{0}<n_{1}<... of integers for which
Define the function f:[1,)R such that f(x)=f([x]).
The required sequence is constructed randomly. Take a random variable t[0,1] of uniform distribution. Set n_{0}=1 and for i=1,2,.... Then
The value of n_{1} is 2,3,4 or 5 if , , , or , respectively. Therefore
and
There always exists a sequence 1=n_{0}<n_{1}<...<n_{k}=N+1 such that
n_{1}a_{n0}+n_{2}a_{n1}+...+n_{k}a_{nk1}<3a_{1}+e(a_{2}+a_{3}+...+a_{N}).
Statistics:
